The solution to the initial value problem y" - 2y' + 10y = 0, y(0) = 0, y'(0) = 6 is y(t) = 6[tex]e^t[/tex] × sin(3t).
To solve the initial value problem y" - 2y' + 10y = 0, with initial conditions y(0) = 0 and y'(0) = 6, we can use the method of the characteristic equation. Let's solve it step by step:
Step 1: Characteristic equation
We assume the solution has the form y = [tex]e^{(rt)[/tex], where r is a constant. Substituting this into the differential equation, we get:
r² - 2r + 10 = 0
Step 2: Solve the characteristic equation
Solving the quadratic equation, we find the roots:
r = (2 ± sqrt(2² - 4(1)(10))) / 2
r = (2 ± sqrt(-36)) / 2
r = 1 ± 3i
Step 3: General solution
Since the roots are complex, the general solution of the differential equation can be written as:
y(t) = [tex]e^{(1t)[/tex] (A × cos(3t) + B × sin(3t))
Step 4: Apply initial conditions
Using the initial condition y(0) = 0, we substitute t = 0 into the general solution:
0 = A × cos(0) + B × sin(0)
0 = A
Using the initial condition y'(0) = 6, we substitute t = 0 into the derivative of the general solution:
6 = (A × 1 × cos(0) - 3B × sin(0))
6 = A
Step 5: Final solution
Now we have A = 0 and B = 6. Substituting these values into the general solution, we obtain the particular solution:
y(t) = [tex]e^t[/tex] × (0 × cos(3t) + 6 × sin(3t))
y(t) = 6[tex]e^t[/tex] × sin(3t)
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please HELP! USE PYTHAGOREAN THEOREM TO FIND RIGHT TRIANGLE SIDE LENGTHS
formula A(2)+B(2)=C(2)
Answer:
x = √ 40
Step-by-step explanation:
6 x 6 = 36 2 x 2 = 4 36 + 4 = 40The integral ſ sin(x - 2) dx is transformed into ', g(t)dt by applying an appropriate change of variable, then g(t) is: g(t) = cos (33) g(t) = sin (5) This option This option g(t) = cos (3 g(t) = sin This option
The integral oſ sin(x - 2) dx is transformed into g(t)dt by applying an appropriate change of variable is g(t) = sin(t).
To transform the integral ∫sin(x - 2) dx into the form ∫g(t) dt using a change of variable, we can let u = x - 2.
Then, differentiating both sides with respect to x gives du = dx.
Substituting these values in the integral, we have:
∫sin(x - 2) dx = ∫sin(u) du
The integral has now been transformed into the integral of sin(u) with respect to u, denoted as g(t) dt.
Therefore, g(t) = sin(t).
So, the correct option is g(t) = sin(t).
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helppppppppppp meeeeeeeeeee
Answer:
It's 11.72
Step-by-step explanation:
Area=)1/2×15/4×25/4
Answer》11.72
Hope it helps...
Have a great day
Answer:
i have made it in above picture
task 1
Find the surface area of the Trumpet.
The surface area of the trumpet is [tex]\( 1256.64 \pi \)[/tex] square feet.
To find the surface area of the trumpet, we need to calculate the areas of the curved surface and the base separately, and then sum them.
The curved surface area of a truncated cone can be calculated using the formula:
[tex]\[ CSA = \pi \times (r_1 + r_2) \times l \][/tex]
Where [tex]\( r_1 \) and \( r_2 \)[/tex] are the radii of the two bases, and [tex]\( l \)[/tex] is the slant height of the truncated cone.
Given that the base diameter is [tex]40[/tex] feet, the radius of the larger base [tex](\( r_1 \))[/tex] is half of that, which is [tex]20[/tex] feet. The slant height [tex](\( l \))[/tex] can be calculated using the Pythagorean theorem:
[tex]\[ l = \sqrt{(h^2 + (r_1 - r_2)^2)} \][/tex]
The height [tex]h[/tex] of the truncated cone is [tex]30[/tex] feet, and the radius of the smaller base [tex](\( r_2 \))[/tex] can be calculated as half the diameter, which is [tex]10[/tex] feet.
Substituting the values into the equations:
[tex]\[ l = \sqrt{(30^2 + (20 - 10)^2)} = \sqrt{(900 + 100)} = \sqrt{1000} = 10\sqrt{10} \]\[ CSA = \pi \times (20 + 10) \times (10\sqrt{10}) = 30\pi\sqrt{10} \][/tex]
The base area of the truncated cone is the area of a circle with radius [tex]\( r_1 \):\[ BA = \pi \times r_1^2 = \pi \times 20^2 = 400\pi \][/tex]
Finally, we can find the total surface area by adding the curved surface area and the base area:
[tex]\[ Surface \, Area = CSA + BA = 30\pi\sqrt{10} + 400\pi \][/tex]
[tex]\[ Surface \, Area = 30\pi\sqrt{10} + 400\pi \]\[ Surface \, Area = \pi(30\sqrt{10} + 400) \]\[ Surface \, Area \approx 1256.637 \pi \]\[ Surface \, Area \approx 1256.64 \pi \][/tex]
Therefore, the simplified surface area of the trumpet is approximately [tex]\( 1256.64 \pi \)[/tex] square feet.
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Find the charge on the capacitor in an LRC-series circuit at t = 0.03 s when L = 0.05 h, R = 3.12, C = 0.008 f, E(t) = 0 V, 9(0) = 4 C, and i(0) = 0 A. (Round your answer to four decimal places.) 0.8630 хс Determine the first time at which the charge on the capacitor is equal to zero. (Round your answer to four decimal places).
At t = 0.03 s, the charge on the capacitor in the LRC-series circuit is 0.8630 C.
In an LRC-series circuit, the charge on the capacitor can be calculated using the formula Q(t) = Q(0) * e^(-t/(RC)), where Q(t) is the charge at time t, Q(0) is the initial charge on the capacitor, R is the resistance, C is the capacitance, and e is the base of the natural logarithm.
Given the values L = 0.05 H, R = 3.12 Ω, C = 0.008 F, E(t) = 0 V, Q(0) = 4 C, and i(0) = 0 A, we can substitute these values into the formula. Using the given time t = 0.03 s, we can calculate the charge on the capacitor.
Plugging in the values, we have Q(0.03) = 4 * e^(-0.03/(3.12*0.008)). Evaluating this expression gives us Q(0.03) ≈ 0.8630 C, rounded to four decimal places.
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HELPPP PLSSSS ASAPP What is the slope of the line below? If necessary, enter your answer as a
fraction in lowest terms, using the slash (/) as the fraction bar. Do not enter
your answer as a decimal number or an equation.
(2,2) (-1,-4)
Answer:
2
Step-by-step explanation:
The formula to find the slope given 2 points is y2-y1/x2-x1. Now lets plug in the numbers. (2,2) is the first set of points and (-1,-4) is the second set of points. So we have -4-2/-1-2. We have -6/-3. Now we must simplify, giving us 2.
1) 1/3(x+3)+1/6=1/2(x-1)-(x-3)
Answer:
x= 8/5
Step-by-step explanation:
Let's solve your equation step-by-step.
1
3
(x+3)+
1
6
=
1
2
(x−1)−(x−3)
Step 1: Simplify both sides of the equation.
1
3
(x+3)+
1
6
=
1
2
(x−1)−(x−3)
1
3
(x+3)+
1
6
=
1
2
(x−1)+−1(x−3)(Distribute the Negative Sign)
1
3
(x+3)+
1
6
=
1
2
(x−1)+−1x+(−1)(−3)
1
3
(x+3)+
1
6
=
1
2
(x−1)+−x+3
(
1
3
)(x)+(
1
3
)(3)+
1
6
=(
1
2
)(x)+(
1
2
)(−1)+−x+3(Distribute)
1
3
x+1+
1
6
=
1
2
x+
−1
2
+−x+3
(
1
3
x)+(1+
1
6
)=(
1
2
x+−x)+(
−1
2
+3)(Combine Like Terms)
1
3
x+
7
6
=
−1
2
x+
5
2
1
3
x+
7
6
=
−1
2
x+
5
2
Step 2: Add 1/2x to both sides.
1
3
x+
7
6
+
1
2
x=
−1
2
x+
5
2
+
1
2
x
5
6
x+
7
6
=
5
2
Step 3: Subtract 7/6 from both sides.
5
6
x+
7
6
−
7
6
=
5
2
−
7
6
5
6
x=
4
3
Step 4: Multiply both sides by 6/5.
(
6
5
)*(
5
6
x)=(
6
5
)*(
4
3
)
x=
8
5
Evaluate : ab−5c+b
for a = 12, b = 3 and c = - 4
Answer:
59
Step-by-step explanation:
[tex](12*3)-5*(-4)+3=59[/tex]
For matrices: A = -1-41] 0 2 2 2 30] B = 33 -1 2 A. What is the dimension of A and B? B. What is the dimension of A B? C. What is A'B?
A. the dimension of A is 3x3. the dimension of B is 3x2.
B. the dimension of AB will be 3x2.
C. The value of AB = [[ -15 14] [ 4 -2] [ 13 -9]]
A. The dimension of a matrix is given by the number of rows and columns it has.
For matrix A, we have:
A = [[-1 -4 1], [0 2 2], [2 3 0]]
The matrix A has 3 rows and 3 columns, so the dimension of A is 3x3.
For matrix B, we have:
B = [[2 0], [3 -3], [-1 2]]
The matrix B has 3 rows and 2 columns, so the dimension of B is 3x2.
B. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In this case, A has 3 columns and B has 3 rows, so we can multiply them.
The dimension of AB will be the number of rows of A and the number of columns of B. Therefore, the dimension of AB will be 3x2.
C. [-1 -4 1] * [2 0]
[0 2 2] [3 -3]
[2 3 0] [-1 2]
[-2-12-1 0+12+2]
= [0+6-2 0-6+4]
[4+9+0 0-9+0]
[ -15 14]
= [ 4 -2]
[ 13 -9]
The value of AB = [[ -15 14] [ 4 -2] [ 13 -9]]
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Given question is incomplete, the complete question is below
For matrices: A =[[ -1 -4 1] [0 2 2][ 2 3 0]] B =[[2 0] [3 -3 ][-1 2] ]
A. What is the dimension of A and B?
B. What is the dimension of A*B?
C. What is A*B?
I NEED HELPPP ASAP pleaseeeeeee
Answer:
Subtract 23 from both sides
Step-by-step explanation:
This is how algebra works, to get a variable by itself you have to do the opposite to both sides. Hope this helped.
Answer:
It's D: subtract 23 from both sides of the equation because inverse operations
Step-by-step explanation:
yo please help its for a grade!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
(1, 2.5 (or 3/2))
Step-by-step explanation:
It's the point the two lines meet
Convert 42 to a base two
numeral
which fraction correctly represents 0.54
Answer:
27/50 is the fraction.
Step-by-step explanation:
A small internet trading company estimates that each network blackout results in a $500 loss. Compute expectation and variance of this company’s daily loss due to blackouts.
x = 0, 1, 2
Px = 0.7, 0.2, 0.1
The expectation (mean) of the company's daily loss due to blackouts is $200, and the variance is $350.
To compute the expectation of the company's daily loss, we multiply each possible loss value by its corresponding probability and sum them up. In this case, we have three possible loss values (0, 1, and 2) with their respective probabilities (0.7, 0.2, and 0.1). Multiplying each loss value by its probability and summing them gives us the expected value of $200.
To calculate the variance, we need to find the squared differences between each possible loss value and the expected value, multiply them by their respective probabilities, and sum them up. Squaring the differences ensures that negative differences do not cancel out positive differences. In this case, the variance is calculated as (0 - 200)^2 * 0.7 + (1 - 200)^2 * 0.2 + (2 - 200)^2 * 0.1, resulting in a variance of $350.
The expectation and variance provide useful measures of the central tendency and variability, respectively, of the company's daily loss due to blackouts.
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The mean of a normal probability distribution is 400 pounds. The standard deviation is 10 pounds.
a. What is the area between 415 pounds and a mean of 400 pounds?
b. What is the area between the mean and 395 pounds?
c. What is the probability of selecting a value at random and discovering that it has a value of less than 395 pounds?
The probability of selecting a value at random and discovering that it has a value of less than 395 pounds is 0.3085.
a. The area between 415 pounds and a mean of 400 pounds is:
0.9332 - 0.5 = 0.4332.
b. The area between the mean and 395 pounds is:
0.5 - 0.3085 = 0.1915
c.The probability of selecting a value at random and discovering that it has a value of less than 395 pounds is 0.3085.
Given the mean of a normal probability distribution as 400 pounds and the standard deviation as 10 pounds.
We have to calculate the area between 415 pounds and a mean of 400 pounds, the area between the mean and 395 pounds, and the probability of selecting a value at random and discovering that it has a value of less than 395 pounds.
a. What is the area between 415 pounds and a mean of 400 pounds?
The Z-score can be calculated as follows:
Z = (x - μ) / σ
= (415 - 400) / 10
= 1.5
From the Z-table, the area to the left of 1.5 is 0.9332.
Therefore, the area between 415 pounds and a mean of 400 pounds is:
0.9332 - 0.5 = 0.4332.
b. What is the area between the mean and 395 pounds?
The Z-score can be calculated as follows:
Z = (x - μ) / σ
= (395 - 400) / 10
= -0.5
From the Z-table, the area to the left of -0.5 is 0.3085.
Therefore, the area between the mean and 395 pounds is:
0.5 - 0.3085 = 0.1915.
c. What is the probability of selecting a value at random and discovering that it has a value of less than 395 pounds?
The Z-score can be calculated as follows:
Z = (x - μ) / σ
= (395 - 400) / 10
= -0.5
From the Z-table, the area to the left of -0.5 is 0.3085.
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if the diameter of a circle is 124 centimeters, then what is the radius.
Answer:
radius = 62 cm
Step-by-step explanation:
radius = diameter/2
radius = 124/2 = 62 cm
Answer:
the radius 62
Step-by-step explanation:
the radius is really just the diameter split in half.
So, 124/2 = 62
Solve the inequality. Express the solution both on the number line and in interval notation. Use exact forms (such as fractions) instead of decimal approximations. 3x-4 a) x²-2x-3≥0 b) 6x-2x² > 0 c); ≤0 9x+17
a) The solution to x²-2x-3≥0 is x ≤ -1 or x ≥ 3, expressed in interval notation as (-∞, -1] ∪ [3, ∞).
b) The solution to 6x-2x² > 0 is x < 0 or x > 3, expressed in interval notation as (-∞, 0) ∪ (3, ∞).
c) The solution to 9x+17 ≤ 0 is x ≤ -17/9, expressed in interval notation as (-∞, -17/9].
a) To solve the inequality x²-2x-3≥0, we can factor the quadratic expression as (x-3)(x+1) ≥ 0. We find that the inequality is satisfied when x ≤ -1 or x ≥ 3. The solution is expressed in interval notation as (-∞, -1] ∪ [3, ∞).
b) To solve the inequality 6x-2x² > 0, we can factor out 2x from the expression to get 2x(3-x) > 0. We find that the inequality is satisfied when x < 0 or x > 3. The solution is expressed in interval notation as (-∞, 0) ∪ (3, ∞).
c) To solve the inequality 9x+17 ≤ 0, we isolate x by subtracting 17 from both sides to get 9x ≤ -17. Dividing both sides by 9, we find x ≤ -17/9. The solution is expressed in interval notation as (-∞, -17/9].
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What do you know about a triangle with an adjacent leg to hypotenuse ratio value of 0.839?
Step-by-step explanation:
Use soh cah toa to solve
it is cos(α)=.839
so the angle of α is 32.9653
Hope that helps :)
Trigonometric identity is cosine. And a is 32.965327740597°.
What are the trigonometric identities?
Equations using trigonometric functions that hold true for all possible values of the variables are known as trigonometric identities.
Given:
A triangle with an adjacent leg to hypotenuse ratio value of 0.839.
Assume the triangle is a right-angled triangle.
In the right-angled triangle,
the ratio of adjacent leg to hypotenuse is cosine.
So,
Cos(a) = Adjacent leg/Hypotenuse
Cos(a) = 0.839
a = Cos⁻¹(0.839)
a = 32.965327740597°
Therefore, angle measure of a is 32.965327740597°.
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The perimeter of a square is 28 cm. Find its area and the length of its diagonal.
help me
Step-by-step explanation:
area: 28/4=7
7x7=49
diagonal=square root 7²+7²
9.89949
9.90
hope it can help you
Juan has a profitable web business of selling T-shirts priced at $25 each. His demand is pretty steady throughout the year (his website is up and running 365 days a year), approximately normally distributed with a mean of 30 T-shirts/day and a standard deviation of 10 T-shirts/day. He has a supplier in China that charges him $5 per T-shirt T and a flat rate of $150 every time he places an order. Orders take exactly 50 days to arrive by container ship. His calculates his annual per unit holding costs at 20% of the wholesale cost of T-shirts. a) What type of inventory management problem is this? Explain your answer. i) Newsvendor (single period) model ii) EOQ model with continuous demand distribution 111) EOQ model with discrete demand distribution b) Calculate how many T-shirts he should order from his China supplier at a time. c) Calculate the level at which he should reorder T-shirts from China to experience at most a 10% chance of a stocking out. a T-shirts, Juan should place an order for d) Fill in: When inventory drops to more T-shirts.
Juan should order approximately 1,813 T-shirts from his China supplier at a time.
How to explain the informationThis inventory management problem can be categorized as the EOQ (Economic Order Quantity) model with continuous demand distribution. In this case, the demand for T-shirts is approximately normally distributed, and the EOQ model assumes a continuous demand pattern.
In order to calculate the optimal order quantity (EOQ), we can use the following formula:
EOQ = ✓((2 * D * S) / H)
where:
D = Annual demand (mean) = 30 T-shirts/day * 365 days/year = 10,950 T-shirts/year
S = Ordering cost per order = $150
H = Holding cost per unit = 20% of wholesale cost = 0.2 * $5 = $1
EOQ = ✓((2 * 10,950 * 150) / 1)
= ✓(3,285,000)
≈ 1,812.99
Therefore, Juan should order approximately 1,813 T-shirts from his China supplier at a time.
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Which sentence is Correct and please take your time !!!
Answer:
b
Step-by-step explanation:
Answer:
The answer is the Last one
Please someone help with this
I’v been stuck on it all day
By the way I have already been sent that link that hacks you
Just show the work
ok, i will help you.
For the first equation, we have points
(-1, -4)
and
(1, -1)
we also know the y-intercept is
(0, -2)
we can make systems of equations to solve for the equation of this exponiental function
y=ab^x
-1=ab
-2=a*1
a=-2
-1=-2(b)
b=1/2
The exponiental fufnction here is y=(-2)(1/2)^x
2nd equation
(0, 6)
(1, 12)
12=ab
6=a*b^0=6
a=6
12=6(b)
b=2
2nd equation is
y=6(2)^x
Help me please with this problem!!
Answer:
139°
Step-by-step explanation:
Corresponding angles theorem:
When a transversal line intersects two parallel lines, the angles formed from the intersection are congruent in pairs. For example with your problem, using the Corresponding Angles Theorem (one of the few theorems/postulates used to find angles on transversals), the angles are paired on the same place on the first parallel line as they are on the second. Since the corresponding pair of (x)° is given, the angle of x is given. (x)° = 139°
I've provided an image courtesy of tutors.com that will help remember corresponding angles for you.
Using the rate of Rs 105 per US Dollar, calculate
the US Dollars for Rs 21000.
Answer:
Hey Aryabd interested to talk with me. Come in comments.
The US Dollars for Rs 21000 would be 200 USD
What is the fundamental principle of multiplication?Multiplication is the mathematical operation that is used to determine the product of two or more numbers. If an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.
We are given that the rate of Rs 105 per US Dollar,
We have to calculate the US Dollars for Rs 21000.
105 Rs = 1 USD
21000 Rs = x USD
Now the proportion can be;
105x =21000
x =21000 /105
x= 200 USD
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Triangle CDE, with vertices C(-8,-7), D(-2,-8) and E(-5,-2), is drawn on the coordinate grid below.
What is the area, in square units, of triangle CDE?
Answer:
Area = 24.75 sqr units
Step-by-step explanation:
You will need these formulas:
[tex]d = \sqrt{(x_2 - x_1)^2 + (y_2-y_1)^2}[/tex]
Midpoint = [tex](\frac{x_{1} + y_{1} }{2} , \frac{x_{2} + y_{2} }{2})[/tex]
Area = b x h
Let us treat CD as the base. Find the length of the base with the distance formula. Use the coordinates for points C & D.
[tex]d = \sqrt{(-2 - (-8))^2 + (-8-(-7))^2}[/tex]
[tex]d = \sqrt{37}[/tex]
The base is [tex]\sqrt{37}[/tex].
The height is the distance between point E and the midpoint of line CD.
Midpoint of CD = [tex](\frac{-8 + (-7) }{2} , \frac{-2 + (-8) }{2})[/tex] = ([tex]-\frac{15}{2}[/tex], [tex]-5[/tex])
Use the distance formula to find the height.
[tex]d = \sqrt{(-5 - (-\frac{15}{2} ))^2 + (-2-(-5))^2}[/tex]
[tex]d = \frac{\sqrt{61} }{2}[/tex]
Find the area with the two distances that were found.
Area = [tex](\sqrt{37}) (\frac{\sqrt{61} }{2})[/tex]
Area = [tex]\frac{\sqrt{2257} }{2}[/tex]
Area = 24.75 sqr units
I NEED HELP
Please write the right answer
Your portfolio must include a minimum of the following five types of equations and solutions:
Two one-step equations
Two equations that contains fractions
One equation with distributive property
One equation with decimals
One real-world problem that is solved by an equation
Remember that each equation must include at least one variable. Once you have created each equation, you will solve it and show your work. Pretend that you are teaching the equations to a new pre-algebra student. Or you can actually teach them to a sibling or friend!
This is a total of 7 equations and solutions.
Answer:
1). (x - 4 = 13)
2). (r + 17 = 51)
3). (7/9x =28)
4). (2/3x = 2)
5). { 3(x-2) = -9 }
6). (x -1.75 = 8.85)
7). Natalie buys organic almonds priced at $77 from the grocery store. How much did
she pay the cashier, if she received $23 in change?
Step-by-step explanation:
1. add 4 on both sides and it will be x=17
2. subtract 17 on both sides and it will be x= 34
3. multiply 9/7 on both sides and it will be x= 257/7 then you divide 257 by 7 then the final answer will be x= 36
4. multiply 3/2 on both sides and it will be x=6/2 then you divide 6 from 2 and the finial answer will be x= 3
5. you will distribute 3 into x and 2 and you will be 3x-6=-9 then you will add 6 on both sides so you will get 3x=3 and then divide 3 on both side so you will get x=1
6. add 1.75 on both sides so x= 10.6
7. the equation will be 23 +x = 77 so first you will subtract 23 on both sides and x will equal 54 so Natalie paid $54
Answer:
i'm taking that test rn
Step-by-step explanation:
A car dealership has 180 cars on their lot. If they increase their inventory by 25%, how many cars will be on the lot?
Answer:
There will be 225 cars on the lot.
A car dealership has 180 cars on their lot. If they increase their inventory be 25%, how many cars will be on the lot?
25/100 * 180
= 5 * 9 = 45
Therefore:
180 + 45 = 225 cars
Answer:
There will be 225 cars on the lot.
Which expression is the result of factoring the expression below by taking out its greatest common factor? 16x^2-8x=? Choose 1 answer: A. 8(2x-1) B. 8(2x^2-x) C. 8x(2x-1) D.8x(2x^2-x)
Answer:
I think it is A
Step-by-step explanation:
What is (-49(49 x 5982+3)^2
Answer:
[tex] - 49 \times (49 \times 5982 + 3) {}^{2} \\ = - 49 \times 85919920641 \\ = - 4,210,076,111,409[/tex]
Let D be the region bounded by the two paraboloids z = 2x2 + 2y2 – 4 and z=5 - x2 - y2 where x > 0 and y > 0. Which of the following triple integral in cylindrical coordinates allows us to evaluate the volume of D? - √3 5-2 Salon dzdrde None of thes
Option D is the correct choice.
To evaluate the volume of D, the triple integral in cylindrical coordinates is required. So, let's derive the required triple integral.
Region D is bounded by two paraboloids z = 2x2 + 2y2 - 4 and z = 5 - x2 - y2 where x > 0 and y > 0.
In cylindrical coordinates,r = √(x^2 + y^2)z = zθ = tan-1(y/x)For the first paraboloid, the cylindrical equation of the paraboloid is: z = 2x2 + 2y2 - 4
By substituting the cylindrical coordinates values in this equation we get z = 2r2 sin2θ + 2r2 cos2θ - 4z = 2r2 (sin2θ + cos2θ) - 4z = 2r2 - 4 Now for the second paraboloid, the cylindrical equation of the paraboloid is: z = 5 - x2 - y2By substituting the cylindrical coordinates values in this equation we get:z = 5 - r2 The limits of r are 0 and √5; and the limits of θ are 0 and π/2.
Finally, the limits of z are obtained by equating the above two paraboloids.2r2 - 4 = 5 - r22r2 + r2 = 9r2 = 3z = 3We have got all the limits of cylindrical coordinates, we can now write the triple integral in cylindrical coordinates which evaluates the volume of D.
The triple integral is:∫(0 to π/2)∫(0 to √5)∫(2r2 - 4 to 3) r dz dr dθ
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